Continuity on a set

A function is continuous on E if it is continuous at every point of E.

Continuity on a set

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces , let $E\subseteq X$ , and let $f:E\to Y$ . The function $f$ is continuous on $E$ if it is continuous at every point of $E$ , i.e.

\forall x_0\in E,\ \forall \varepsilon>0,\ \exists \delta>0\ \text{such that}\ \forall x\in E,\ \bigl(d_X(x,x_0)<\delta \Rightarrow d_Y(f(x),f(x_0))<\varepsilon\bigr).

Continuity “on a set” is the standard notion for stating global theorems (e.g., continuous images of compact sets are compact). See also uniform continuity .

Examples:

$f(x)=\sin x$ is continuous on $\mathbb{R}$ .
$f(x)=1/x$ is continuous on $(0,\infty)$ and on $(-\infty,0)$ , but not continuous on $\mathbb{R}$ as a whole since $0$ is not in its domain.
The function $f(x)=\sqrt{x}$ is continuous on $[0,\infty)$ .